integrate_double_polar: Numerical Double Integration in Polar Coordinates

Description

Calculates the definite double integral of a function f(x, y) over a polar region R, defined by constant limits for theta and functional limits for the radius r, R: \(g1(theta) <= r <= g2(theta)\), \(alpha <= theta <= beta\). The integration order used is r dr dtheta. Uses the Composite Simpson's Rule for numerical approximation.

Usage

integrate_double_polar(
  f,
  theta_min,
  theta_max,
  r_limit1,
  r_limit2,
  n_theta = 100,
  n_r = 100,
  plot_domain = TRUE
)

Value

A list containing:

integral_value: The calculated approximation of the integral.
domain_plot: The ggplot2 object representing the domain (if plot_domain = TRUE).

Arguments

f: A function R of two variables, f(x, y), returning a numeric value (the original integrand).
theta_min: The constant lower limit for the outer integral (alpha).
theta_max: The constant upper limit for the outer integral (beta).
r_limit1: A function R of one variable defining the inner integral's lower limit (r = g1(theta)).
r_limit2: A function R of one variable defining the inner integral's upper limit (r = g2(theta)).
n_theta: Number of subintervals for the outer integration (theta). Must be even. Default is 100.
n_r: Number of subintervals for the inner integration (r). Must be even. Default is 100.
plot_domain: Logical. If TRUE, generates a ggplot2 plot of the integration domain in the Cartesian plane. Default is TRUE.